Papers

It is known that the study of vibration characteristics at the tip of the micro-cantilever and its relationship to the sample plays a very important role in improving the resolution of an Atomic Force Microscopy (AFM). In this paper, within the Euler Bernoulli beam model, a mathematical model, defined as a model with two unknown inputs and two measured outputs, is considered for the simultaneous determination of the unknown bending moment and the shear force at the tip of the micro-cantilever from two feasible measured outputs at the same tip: the deflection and the slope. This model leads to the following inverse problem: find $M(t)$ and $g(t)$ in $\rho_A(x)u_{tt}+\mu(x) u_{t}+(r(x)u_{xx})_{xx}=0$, $(x,t)\in \Omega_T:=(0,\ell)\times (0,T)$ subject to the boundary conditions $u(0,t)=u_{x}(0,t)=0$, $\left(r(x)u_{xx}\right)_{x=\ell}=M(t)$, $\left((r(x)u_{xx})_x\right)_{x=\ell}=g(t)$, and the homogenous initial conditions, from the measured outputs $w_{\ell}(t):=u(\ell,t)$ and $\theta_{\ell}(t):=u_x(\ell,t)$. It is proved that the vector-form input-output map $\mathcal{P}:=\left (\Phi, \Psi \right )$, with $\left (\Phi q \right )(t):=u(0,t;q)$ and $\left (\Psi q\right )(t):=u_x(0,t;q)$, where $q(t):=\left (M(t),g(t)\right )$, corresponding to the inverse problem, is compact and Lipschitz continuous. This result allows us to prove the existence of a solution of the minimization problem for the Tikhonov functional. As a consequence, the existence of a quasi-solution to the inverse problem is established. Furthermore, a vector form expression for the Frechet gradient of the Tikhonov functional is derived, and the Lipschitz continuity of the Frechet gradient is rigorously proven. This crucial property ensures the monotonic behavior of iterative gradient-based numerical methods.

In this work, we investigate a chemotaxis-driven fish–mussel ecosystem model described by a coupled system of partial differential equations subject to homogeneous Neumann boundary conditions. Under suitable assumptions on the system parameters and initial data, we establish the global existence of classical solutions by employing semigroup methods together with a priori estimates. We also examine the possible blow-up behavior of solutions in a three-dimensional domain. To support the theoretical analysis, a finite element method is developed for the numerical approximation of the system, and convergence studies based on mesh refinement are carried out to verify the accuracy and stability of the proposed numerical scheme. Furthermore, numerical simulations illustrating the blow-up behavior of solutions in the computational domain are presented.

In this paper, we study $(-1)$-homogeneous steady solutions to the Euler equations on $\mathbb{R}^n \setminus \{0\}$. In low dimensions $n=2,3$, such flows are known to be essentially trivial. In contrast, we show that in higher dimensions $n \ge 4$, every $(-1)$-homogeneous Euler flow is a geodesible vector field with constant Bernoulli function. Moreover, any $(-1)$-homogeneous geodesible field is induced by a geodesible field on the sphere $\mathbb{S}^{n-1}$. In particular, in the case $n=4$, every $(-1)$-homogeneous Euler flow is obtained as an extension of a Beltrami field on $\mathbb{S}^{3}$.

We investigate the global well-posedness of a class of generalized Hillen–Painter systems – specifically, supercritical volume-filling chemotaxis models – in $\mathbb{R}^2$ under the influence of Couette flow. It is well established that, in the absence of fluid flow, solutions to this system may develop finite-time singularities (blow-up) for arbitrary initial cell mass. It is proved that the introduction of a Couette flow with sufficiently large amplitude guarantees the global existence of solutions for all initial masses. By employing a novel frequency decomposition technique, we successfully remove the mass threshold limitation presented in previous studies on the domain $\mathbb{T}\times\mathbb{R}$ (Wang et al., Commun. Contemp. Math.), thereby establishing global regularity in the whole space without any smallness assumptions.

We propose a mathematical framework for redistricting compactness grounded in the classical soap bubble problem: the partition of a planar region into N subregions of prescribed population that minimizes total shared boundary length. This formulation, which we call the soap bubble model, yields a physically and mathematically natural optimality criterion that generalizes all existing per-district compactness measures. We carefully distinguish two formulations: minimizing physical boundary length in geographic coordinates, and minimizing a density-weighted length in the coordinates produced by the Monge–Ampere density-flattening map. In the transformed coordinates, the first-order conditions for a minimizer are precisely the classical two-dimensional Plateau's laws – circular arc boundaries, 120-degree triple junctions, and perpendicular boundary contact. In geographic coordinates, the Euler–Lagrange equation shows arc curvature varies proportionally with local population density, so boundaries curve more tightly through dense areas and gently through sparse ones. Remarkably, the 120-degree junction condition and 90-degree boundary condition are unchanged by the density, as they follow from a force-balance argument that does not involve density. We adopt the physical-length objective as the natural redistricting criterion – it minimizes actual miles of district boundary and requires no coordinate transformation – and derive the corresponding density-modified volume-preserving mean curvature flow as its gradient descent. The N-state Potts model on the census block adjacency graph with population-weighted pressure updates is the natural discrete implementation of this flow. We define a multiplicative gerrymandering metric decomposed into tactical and systemic components and discuss practical computation, including grid-pinning effects in the discrete setting.

We investigate kink antikink collisions in higher order scalar field theories described by the higher order models and their shifted periodic extensions. Both classes of models possess three degenerate vacuum states and support topological kink solutions with asymmetric profiles and algebraically decaying tails. By extending conventional polynomial potentials across multiple spatial sectors, we construct shifted periodic high order field theories and examine how this modification affects the scattering dynamics of topological defects. The primary objective of this study is to provide a comparative numerical analysis of kink collisions in the standard and shifted periodic versions of these higher order models. Using direct numerical simulations, we determine the critical velocities that separate capture from escape regimes and identify resonance structures associated with energy exchange between translational and internal vibrational degrees of freedom. Particular attention is devoted to the emergence of escape windows, quasi-fractal patterns, and the role of algebraic tails in shaping the collision outcomes. Our results demonstrate that, although the conventional and shifted periodic models exhibit similar kink antikink configurations, important quantitative differences arise in their critical velocities, resonance structures, and scattering characteristics. The findings further confirm that both classes of models support resonant energy transfer mechanisms analogous to those observed in lower order theories, while simultaneously exhibiting novel features associated with higher-order interactions and long range effects. These results contribute to the growing understanding of nonlinear excitations in scalar field theories and provide new insights into the dynamics of topological solitons in shifted periodic systems

We prove a fixed finite-window structural theorem for sharp localized Navier-Stokes packages, formulated as both a local-to-clean detection theorem and an anti-phantom principle. The result addresses whether a defect visible in the baseline quotient geometry can disappear after pressure-tail enrichment, residual transfer, quotienting, and clean-to-local detector comparison. Under synchronized representatives, baseline-to-tail visibility, component comparison, residual-ledger closure, detector comparison, chart visibility, and a clean quotient gap, any baseline-visible defect is either detected by the localized detector or charged to an explicit quotient-residual ledger. Quantitatively, M_Lambda^loc(D-zeta_*) >= c_{Lambda,0} Dist_{loc,intg,0}(D,Gamma^{intg}*{Lambda,adm}) - E^{quot}*{Lambda,0}(D). The proof assembles three modules: pressure-tail visibility, componentwise residual-ledger closure, and detector comparison. The anti-phantom interpretation is that a baseline-visible defect cannot be simultaneously detector-silent and residual-cheap. We also record provenance for the imported quotient interface, finite-dimensional pressure-tail models, explicit matrix realizations of the structural inputs, NS-generated coordinate realizability, compactness criteria for clean pressure images, and reduced pressure/tax kernel-free criteria.

We study the propagation of coupled surface and internal equatorial internal waves. A model of two vertically stratified fluid layers with different constant densities is employed. Taking Coriolis forces into account, we derive explicit solutions to the linearized governing equations which assumes irrotational fluid motion in both layers separately, and further obtain the dispersion relation which determines the phase speeds of propagating surface and internal waves. We prove a result on solutions to the dispersion relations which greatly simplifies our subsequent analysis of the nonlinear dynamical systems which describe the motion of the fluid in the upper layer. Phase portraits for all possible streamlines in both fluid layers are presented, while furthermore a Lagrangian description of the fluid flow is obtained, and the particle trajectories of the fluid particles are determined.

We establish uniqueness and stability results for a class of perturbations of an unbounded potential in the two-dimensional Schrödinger equation, from the corresponding Dirichlet-to-Neumann map. We assume that the difference between the potentials has a separated product structure. Our proof relies on a specific Carleman inequality.

We study infinite-horizon optimal stopping for normally reflected two-dimensional diffusions in the positive quadrant with max payoff \(G(x_1,x_2)=x_1\vee\alpha x_2\). The non-smooth payoff produces a singular stopping-gain measure on the kink set \(\Delta=\{x_1=\alpha x_2\}\). We prove $\displaystyle \Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\,\sigma_\Delta(dx)$, with $n=(1,-\alpha)$, so the diagonal component is non-positive and strictly negative under local ellipticity. This implies that every interior kink point lies in the continuation region. We further show that the correct value representation uses the resolvent killed at first entry into the stopping set, $\displaystyle V=G-R_r^{\mathcal C}\Gamma$, and give a closed-form reflected Brownian counter-example showing that the unrestricted reflected resolvent is generally wrong. A reflected Brownian benchmark and numerical experiments illustrate the local-time, resolvent-gap, and diagonal-avoidance mechanisms.

We study two inverse problems for a nonlinear dynamical Schrödinger equation with time-dependent magnetic and electric potentials. Under suitable analyticity assumptions, we show that the associated Dirichlet-to-Neumann map uniquely determines the linear magnetic potential and all coefficients of the nonlinear electric potential. We establish both full-data and partial-data uniqueness results. For the partial data problem, assuming that the coefficients are known in a neighborhood of the boundary, uniqueness is obtained using measurements made on arbitrarily small open subsets of the boundary. In addition, we establish the well-posedness of the forward problem.

The Beltrami-Vekua normal form assigns to every smooth first-order real planar elliptic system a complex equation $w_{\bar z}-\mu w_z+\mathcal{A}w+\mathcal{B}\bar w=\mathcal{F}$ by an explicit pipeline. A companion paper showed that the density $\Theta=|\mathcal{B}|^2/(1-|\mu|^2)\,dx\,dy$ and its total mass are invariants under multiplicative gauges $w\mapsto\phi w$ and orientation-preserving diffeomorphisms. The real system carries a larger symmetry: its unknowns may be recombined by any pointwise invertible real-linear substitution $w=\varphi v'+\psi\bar v'$, the complex gauges being the case $\psi\equiv0$. We prove the absorption theorem: re-normalizing through the pipeline after any such substitution returns to the gauge orbit of the original equation, with a universal explicit gauge $\tilde\varphi=-i\lambda/(\varphi-\psi)$, where $\lambda$ is the spectral root of the structure polynomial.

We prove a Trudinger-Moser type inequality in fractional Sobolev spaces with singularities on smooth compact sets of codimension $k$, where $1 < k < d$ and $sp = d$. The singular term is given by the inverse $d$-th power of the distance to the submanifold. The proof is based on a fractional Hardy inequality adapted to smooth submanifolds, and we show the sharpness of the constant. We also establish the equivalence of two natural fractional Sobolev spaces vanishing on the singular set.

We study the global-in-time continuation, past the singularity, of the smooth, non-isentropic, radially symmetric imploding solutions of the compressible Euler equations recently constructed by Chen, Shkoller, and Vicol. In three space dimensions, for all physically relevant adiabatic exponents $\gamma>1$, we consider the Euler solution that evolves smoothly until an implosion singularity forms at the origin at time $t=0$. We then prove that this solution can be uniquely continued for $t>0$ as a reflected outward-propagating shock, sometimes called a reflected blast wave. For $t>0$, the continuation is a globally forward self-similar weak solution of the Euler equations, selected by the Rankine–Hugoniot conditions and the Lax entropy inequality; it is smooth away from the expanding shock sphere and the spatial origin. The structure at the center of symmetry distinguishes these explosions from the classical Guderley reflected shock. In Guderley's continuation, the reflected blast wave leaves a point vacuum at the origin, where the density vanishes. The solutions constructed here exhibit the opposite behavior: for every fixed $t>0$ the density is unbounded at $r=0$ (though it remains locally integrable), while the pressure stays bounded and the temperature vanishes there.

We compute explicitly the infimum of the norms of $W^{1,2}$-extension operators for planar sectors and exhibit an extension operator attaining this infimum. This solves an open problem posed by Maz'ya.

Inspired by \cite{CCP}, we investigate a delayed leader-follower Cucker-Smale model describing the collective dynamics of interacting agents subject to communication lags. We first study the particle dynamics and establish sufficient conditions ensuring the emergence of asymptotic flocking. Our analysis shows that velocity alignment and bounded spatial dispersion persist despite the presence of delays and heterogeneous interactions between leaders and followers. We then derive and analyze two continuum descriptions of the system. In the first regime, the number of leaders is kept fixed while the number of followers tends to infinity, leading to a hybrid particle-kinetic model. In the second regime, both populations become infinitely large, yielding a fully kinetic delayed leader-follower model. For both mean-field formulations, we prove global existence, uniqueness, and Wasserstein stability of measure-valued solutions. These results provide a rigorous mathematical framework for the study of collective dynamics with leadership and memory effects and establish a bridge between delayed flocking models and their continuum counterparts.

Whether the multi-dimensional Euler-Poisson system admits global smooth solutions remains a challenging open problem. In this paper, we construct a class of large-data global smooth solutions to the one-fluid Euler-Poisson system in $\mathbb{R}^d$ ($1\leq d\leq 5$) by using the relaxation dissipation mechanism. Precisely, assuming that the initial density is far from vacuum and $\varepsilon E_0$ is sufficiently small, where $E_0$ denotes the initial energy and $\varepsilon$ is the relaxation time, we establish the global well-posedness of smooth solutions to the Cauchy problem. In particular, the size of the initial perturbation may be arbitrarily large, provided that the relaxation time is sufficiently small. Furthermore, we introduce an effective unknown motivated by Darcy's law to derive quantitative error estimates at the rate $\mathcal O(e^{-\lambda t}\varepsilon)$ between the rescaled Euler-Poisson system and the limiting drift-diffusion system for ill-prepared data. The new ingredient lies in developing the maximum principle for the nonlinear drift-diffusion system with nonlocal effect, which leads to the large-data global existence.

We consider the Neumann initial boundary value problem associated to the chemotaxis system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \big((u+1)^{m-1} u_x - u(u+1)^m v_x\big)_x & \text{in $(0, 1) \times (0, \infty)$}, \\ v_t = v_{xx} - v + u, &\text{in $(0, 1) \times (0, \infty)$}, \end{cases} \end{align} where $m \in \mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all $m \in (-\infty, -1] \cup (0, \infty)$ and even for all $m \in \mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \hookrightarrow L^\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\delta$ of the heterogeneities is much smaller (microscopic) than the interface width $\epsilon$ (mesoscopic). In a companion paper, we show that if the ratio $\epsilon^{-1} \delta$ vanishes fast enough as $\epsilon \to 0$, then the functionals converge to an effective surface energy where the energy density is determined by homogenization effects originating at microscopic scales. Here we prove that if the ratio $\epsilon^{-1} \delta $ vanishes too slowly, the limit of the functional may actually be smaller than this homogenized energy. We refer to this as the rare events regime. In the case of the random checkerboard in dimension one, we use large deviations techniques to give a complete description of the rare events regime, showing that the limiting energy depends in a nontrivial way on the limit of $\epsilon^{-1} \delta | \log \epsilon |$. We further construct, in any dimension, examples of random media in which rare events become relevant at algebraic scales $\delta \approx \epsilon^{1 + \alpha}$ for an arbitrary $\alpha > 0$, as well as almost periodic examples in which atypical configurations play the same role as rare events.

Collective migration of heterogeneous cell populations is central to many biological and physiological processes, including development and immune response. Recent experimental and theoretical advances have shown how asymmetric interactions with self-generated chemical gradients shape the spatial distribution of distinct cell types within migrating collectives. However, the principles governing robust spatial organisation of heterogeneous cell populations remain poorly understood. Here, we use asymptotic analysis to systematically derive a nonlinear analytical theory for heterogeneous cell collectives guided by self-generated chemotaxis. Our theory disentangles how heterogeneity in cell diffusivity, chemoattractant consumption, and chemotactic sensitivity shape the density profiles of migrating heterogeneous collectives, revealing four distinct dynamical behaviours that together capture all possible regimes. We calibrate our framework to experimental data on the co-migration of dendritic and T cells. We predict that this system operates in a parameter regime that balances intercellular mixing with T-cell localisation at the leading front of the migrating collective. Our theory reveals that this behaviour is enabled by intermediate long-range chemoattractant signalling generated through strong chemoattractant consumption by dendritic cells. Overall, our framework provides general principles for understanding how non-reciprocal chemical interactions shape robust collective migration in heterogeneous cell populations.

In this paper, we study the global dynamics of the 3D ionic Euler-Poisson equations with the parameter of Mach number $\varepsilon$. We first establish the global well-posedness and scattering for the high Mach number regime $0<\varepsilon\leq1$ and pressureless case $\varepsilon=0$. Moreover, we prove the high Mach number limit, showing that the profile of the solution for ionic Euler-Poisson equations converged to that of the pressureless equation as $\varepsilon\rightarrow0$. Our approach combines energy estimates, dispersive estimates and the normal form method. The major difficulty lies in establishing the uniform estimates with respect to the parameter, as the dispersive or resonance structure degenerates when $\varepsilon$ tends to 0. A crucial observation is that despite the disappearance of the pressure ($\varepsilon\rightarrow0$), dispersive phase function always remains a wave-type structure in zero frequencies, which enables us to derive linear and bilinear multiplier estimates adapted to the uniformity of Mach number parameter.

This paper initiates a mathematical investigation of a PDE model for the transport of high voltage direct current via overhead lines. We prove the existence of infinitely many solutions, give necessary conditions for existence, explicitly compute the continuum of all radial solutions, and develop a new numerical algorithm for this problem.

We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \[ \begin{cases} -\Delta u - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n,\cr u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q\in(1,2)$ and $n\geq 2$. A distinctive feature of this problem is the non-Lipschitz singularity of the nonlinearity at the origin, which gives rise to compactly supported ground states and bound states. Using a shooting argument together with a detailed analysis of the linearized variation with respect to the initial value, we prove that for every prescribed integer $k\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros, up to sign reflection and spatial translation. In addition, our analysis yields a classification of radial solutions according to the initial value and describes their behavior near the finite support boundary.

In this work we establish that the recently introduced fractional Sobolev spaces based on the Riesz fractional gradient of Musielak-Orlicz functions by one of the authors, coincide with the space of Bessel potentials of functions on such generalized Orlicz setting. Moreover, we identify them as complex interpolation spaces, and exploiting the well known properties for interpolation of operators we obtain several structural properties for those spaces.

We develop an ABP approach to Sobolev and Michael-Simon type inequalities under volume noncollapsing assumptions. The main new observation is a refinement of Brendle's contact-set argument: the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential, with radius equal to the ABP parameter. This allows one to use lower bounds for the volumes of geodesic balls, either at a fixed scale or under prescribed volume-growth assumptions, rather than positive asymptotic volume ratio. The central application is a Michael-Simon type inequality for immersed submanifolds of ambient manifolds with nonnegative sectional curvature and volume noncollapsing. The resulting inequality contains a lower-order term determined by the noncollapsing scale and applies to submanifolds with controlled mean curvature. In the intrinsic case, the same method gives an ABP proof of Varopoulos' $L^{1}$-Sobolev inequality with lower-order term, identifying the optimal constant in front of the gradient term, as well as explicit lower bounds for the isoperimetric profile in terms of lower bounds on the volumes of geodesic balls. Further geometric applications include Topping-type diameter estimates for submanifolds involving the $L^{n-1}$-norm of the mean curvature and various heat kernel and spectral estimates in the minimal case.

In kinetic theory, interactions between particles are typically assumed to be "all-to-all", meaning that any pair of randomly selected particles may, in principle, interact. This assumption originates from the theory of colliding gas molecules; however, it may be less appropriate for describing other forms of interaction, such as social interactions. These are more naturally characterised as "some-to-some", reflecting the existence of preferential connections between agents. In this paper, we consider homogeneous Boltzmann-type equations on finite graphs that model such networks of preferential interactions, and we rigorously derive their dense graph limit as the number of agents tends to infinity. We also investigate the long-time behaviour of the limiting equation in the case of linear pairwise interactions, characterising the emergent equilibrium distributions and relating them to their counterparts in the classical "all-to-all" setting.

We study the vanishing viscosity limit for the incompressible Navier-Stokes equations with anisotropic viscosity in bounded domains, analyzing certain classes of symmetric flows: plane parallel, pipe parallel and circularly symmetric. By anisotropic viscosity, it is meant here that the viscosity coefficient in the direction normal to the wall is different than that in the direction tangential to the wall. Using boundary layer theory and semigroup techniques, we establish the validity of the vanishing viscosity limit in the energy norm, that is, in $L^2$ in space uniformly in time, for all three classes of flows, with explicit convergence rates. We further obtain higher-order estimates under suitable assumptions on the anisotropic viscosity coefficients. In particular, we consider both the case in which the tangential viscosity coefficient goes to zero faster than the normal one and, conversely, the case when the normal coefficient vanishes faster then the tangential one. Our results extend previous works on isotropic viscosity and provide new examples where the vanishing viscosity limit can be rigorously justified in the anisotropic setting.

We investigate the impact of large synaptic delays on the emergence of periodic dynamics in inhibitory neuronal networks, within the framework of the NNLIF model. Inspired by the work of [11] where the notion of pseudo-equilibria was introduced and developed, and by our earlier analysis in [14], we show that, as the delay tends to infinity, solutions of sufficitently inhibitory networks oscillate between distinct pseudo-equilibria over any finite time interval. Employing the Doeblin-Harris method, we rigorously establish a local convergence in the Cesàro mean toward a limit function determined solely by these pseudo-equilibria.

Many fractals $G$ admit a harmonic immersion into $\R^n$, i.e. an immersion which minimises a natural energy under fixed boundary conditions; we look for harmonic immersions of the Sierpinski gasket into the hyperbolic plane. We show that, given any three points $\tilde A$, $\tilde B$, $\tilde C$ in the hyperbolic plane there is a harmonic map bringing the three points $A$, $B$, $C$ of the boundary of the gasket to $\tilde A$, $\tilde B$, $\tilde C$ respectively. Moreover, if the points $\tilde A$, $\tilde B$, $\tilde C$ are sufficiently close in the hyperbolic distance, then the harmonic map is unique and depends differentiably on $\tilde A$, $\tilde B$, $\tilde C$. Lastly, we show that, if the harmonic map $\phi$ is injective, then it brings geodesics of the gasket $G$ into geodesics of $\phi(G)$.

In this paper, we develop a framework based on differential forms that enables us to deal with the Hall problem on domains in any dimension $n\ge2$. In the case of smooth bounded domains of $R^n$, we prove local existence of mild solutions in subcritical spaces.